Invited Speakers:

Jesús Fernández-Sanchez, Polytechnic University of Catalonia

Alex Fink, Queen Mary University, London

Eugenie Hunsicker, University of Loughborough

Natalia Iyudu, University of Edinburgh

Marianne Johnson, University of Manchester

Paolo Tripoli, University of Nottingham

Date:

14-15 June 2018

Location:

Room 110, School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ

Schedule

Thursday 14th June 2018

10:30-11:00 Coffee/tea/juice

11:00-12:00 Natalia Iyudu, University of Edinburgh

12:00-1:00 Jesús Fernández-Sanchez, Polytechnic University of Catalonia

1:00-2:00 Lunch & discussions

2:00-3:00 Eugenie Hunsicker, University of Loughborough

3:00-4:00 Paolo Tripoli, University of Nottingham

6:00 Dinner

Friday 15th June 2018

9:30-10:00 Coffee/tea/juice

10:00-11:00 Marianne Johnson, University of Manchester

11:00-12:00 Alex Fink, Queen Mary University, London

12:00-12:30 Coffee/tea/juice

Abstracts

Jesús Fernández-Sanchez: On the embedding problem for Markov matrices

The difference between continuous time and discrete time approach is crucial in the design of models for nucleotide substitution. In genomics and phylogenetics, we usually find the continuous time approach as the one preferred to model evolution of DNA as it offers the possibility of constructing models mathematically tractable. This leads to the problem of characterizing those stochastic matrices that can be expressed as the exponential of some rate matrix. This is known in literature as the "Embedding problem for Markov matrices" (Elfving, 1937).

In this talk, we will review some facts about the connection between stochastic and rate matrices and present some new results related to some popular evolutionary models, mainly the Kimura model with 3 parameters (1981) and its submodels. As a consequence, we will be able to compare the volume of embeddable Markov matrices relative to the volume of all Markov matrices within this model. On the other hand, we will present examples showing that mutation rates may not be identifiable from substitution probabilities. These examples also illustrate that symmetries between mutation probabilities do not necessarily arise from symmetries between the corresponding mutation rates.

Alex Fink: Schubert in Gaussian conditional independence

We study conditional independence of sets of coordinates in a multivariate Gaussian distribution, with an interest in the different conditional independence models that are possible. In one natural case, corresponding to certain memoryless processes that generalise Markov chains, a complete answer can be obtained using the tools of Schubert varieties and determinantal ideals. This is joint work with Jenna Rajchgot and Seth Sullivant.

Eugenie Hunsicker: Jumping across the chasm: transitioning from pure mathematics to applied statistics

About four years ago, I had a midlife crisis and decided to change my research area from pure mathematics (geometric analysis and topology) to applied statistics. As a midlife crisis, it was arguably pretty mild (my kids were hoping for a sports car), but it has definitely been an interesting and very educational experience. In this talk, I will talk about how my perspective on applied work has changed as I have begun to work with data producers. I will also talk about where I think there will be interesting mathematical questions in the future, and some of the practical obstructions we need to get past before new mathematics will play a part.

Natalia Iyudu: Applications of Gröbner bases.

I will discuss the Gröbner bases theory and its application to calculation of Hilbert series and related invariants. Examples where this technique can be used and on this bases, for example, Koszulity proved, will be given. It can be applied to the persistent homology problems of data analysis via calculations in Grassmann algebra, to the problems in neural networks, etc.

The Gröbner technique itself originated in computer science, more precisely, in computer algebra, but as we will see, after proper algebraic formulation it can serve for proving some structural and homological results about algebras presented by generators and relations.

Marianne Johnson: Upper triangular tropical matrix identities

Matrices over the tropical semifield arise in models of discrete event systems, optimisation and scheduling problems. In contrast to the case of upper triangular matrices over an infinite field, it is known that the semigroup UT_n(T) of all upper triangular matrices over the tropical semifield satisfy non-trivial semigroup identities. For example, Izhakian and Margolis have shown that the identity:

ABBA AB ABBA = ABBA BA ABBA

holds for all A, B \in UT_2(T). In this talk I will discuss the following questions, relating to some joint work with Laure Daviaud, Mark Kambites, and Ngoc Mai Tran:

* Given a pair of words over a fixed alphabet {A, B}, how to decide if they form a semigroup identity for UT_n(T)?

* Given a single word over a fixed alphabet {A, B}, how to decide if there exists another word with which it forms a semigroup identity for UT_n(T)?

In the case n=2, the answers to these questions can be readily expressed in terms of certain lattice polytope computations. (This case is of particular interest, because the identities satisfied by UT_2(T) turn out to be precisely those satisfied by the bicyclic monoid.

Paolo Tripoli: Configurations of vortices via invariant theory

The study of configurations of vortices in a plane is a well-studied problem in Physics. The case of stationary configurations can be reduced to solving a system of polynomial equations, however, standard computational tools are unable to handle the problem due to its complexity. In an ongoing joint work with Emilie Dufresne, Heather Harrington, and Panos Kevrekidis, we are exploiting the invariance of the system to reformulate the problem in a more approachable way.

Visitor information

• How to get to the University of Glasgow
• Campus Map - We are located at C3
• How to get to the Mathematics & Statistics Building

Local organiser

Dimitra Kosta, University of Glasgow

Sponsors:

We are grateful for the financial support from the Glasgow Mathematical Journal Learning and Research Support Fund, from the Edinburgh Mathematical Society, the London Mathematical Society and the University of Glasgow.